BULLETIN (New Series) OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 37, Number 2, Pages 141–154
S 0273-0979(00)00857-0
Article electronically published on January 21, 2000
VARIATIONS ON CONSERVATION LAWS
FOR THE WAVE EQUATION
CATHLEEN SYNGE MORAWETZ
Abstract. The first part of this paper, presented as an Emmy Noether lecture
in connection with the ICM in Berlin in August 1998, gives some examples of
using Noether’s theorem for conservation laws for Tricomi-like equations and
for the wave equation. It is also shown that equations which are semilinear
variations of the wave equation can very often be handled similarly. The type
of estimate obtained can even be used to get otherwise unobtainable local
estimates for regularity.
The fourth part is an introduction to the relation of black holes to the wave
equation mainly showing the results of D. Christodoulou. His results use much
more difficult estimates not corresponding at all to those in the first part of
the paper.
1. Introduction
It is a very, very great honor to give the Emmy Noether lecture here in Berlin at
the time of the International Congress of 1998, the last of this century. I know that
this honor to women would have been impossible forty years ago. To the best of
my knowledge sometimes the ICM gave special attention to small countries but not
to women or certain minorities. Also my great hope is that say thirty years from
now the need for such a lecture will have evaporated. But now I wish to thank my
sponsors, the Association for Women in Mathematics and the European Women in
Mathematics, as well as the organizers of the Congress.
Mathematics is the queen of the sciences, and Emmy Noether is the queen of
mathematics. I never met her, but she was a good friend of my mentors, Kurt
Friedrichs and Richard Courant, who spoke to me about her many times. A nonreligious Jewish refugee from Nazi Germany, she died at the age of fifty-four in the
USA. She did not live to receive either the job or the accolades that she deserved
for her profound work in developing algebra. On the other hand, she did not live
to know of the horrors that awaited the Jews of Europe, including many of her
old friends; or the terrors of Stalinism, into which her brother disappeared; or the
appalling destruction of the cities of Europe and of many of the friends of her youth.
Here I would like to make use of another of her contributions to mathematics,
this one not to algebra but to partial differential equations. According to what I
have heard, it was not one that she particularly liked. It followed on some work
on invariants that had involved some exceedingly tedious calculations. But it has
generated a number of important physical applications.
Received by the editors July 1, 1999, and in revised form October 6, 1999.
2000 Mathematics Subject Classification. Primary 35Lxx, 35Mxx, 35Qxx, 83Cxx.
Key words and phrases. Nonlinear, semilinear, p.d.e., Tricomi equation, black hole.
c
2000
American Mathematical Society
141
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CATHLEEN SYNGE MORAWETZ
2. Noether’s theorem and the Tricomi equation
Suppose we have a system of equations derivable from a Lagrangian. Noether’s
theorem [16] for conservation laws states that:
To every change of variables (independent or dependent) that leaves the action
of a system—that is, the integral of the Lagrangian—invariant there corresponds a
conservation law for the system.
In other words, to the group of invariant transformations there corresponds a
group of conservation laws.
As a first example we will derive Noether’s theorem for Tricomi-like equations,
that is, for
(2.1)
K(y) uxx + uyy = 0 where
yK(y) ≥ 0,
in a domain Ω that is for the time being arbitrary. This equation behaves like
the wave equation (hyperbolic) for y < 0 and like Laplace’s equation (elliptic) for
y > 0. The action of the Lagrangian is, not surprisingly,
Z
Ku2x + u2y dx dy.
A=
Ω
It is invariant under a shift in x.
Let us compute the first variation, which must vanish,
Z
(2Kuxδux + 2uy δuy ) dx dy +
0 = δA =
Ω
Z
Ku2x + u2y δn ds
∂Ω
where δn is the variation in the normal, δu in u. Integrating the first term by parts,
we find:
Z
2 (Kuxx + uyy ) δu dx dy
0=−
Ω
Z
2(Kux dy − uy dx) δu + (Ku2x + u2y ) δn ds.
+
∂Ω
The first term must vanish since δu is arbitrary. This gives us as expected the
differential equation (2.1). We use next the invariance with respect to x, on the
boundary term,
x → x + ,
y → y,
u(x, y) → u(x − , y),
or δu = −ux , and δn ds = dy. The vanishing of the variation of the action
reduces to
Z
−2 (Kux dy − uy dx) ux + Ku2x + u2y dy
0=
∂Ω
for arbitrary ∂Ω. Or,
Z
0=
−2ux uy dx + Ku2x − u2y dy.
∂Ω
Since ∂Ω is an arbitrary curve, this is the conservation law. Note that the Noether
conservation law “comes out of the boundary” as it does generally. Like every 2D
conservation law it implies the existence of a function U defined by
Ux = −2ux uy ,
Uy = K(y) u2x − u2y .
VARIATIONS ON CONSERVATION LAWS
143
The function U turns out to be very useful for uniqueness theorems for a wide range
of domains that straddle the line y = 0, where the equation changes type.
U has three important properties:
(A) If u = 0 on a curve Γ, then
dy dU ≥ 0 on Γ where y ≥ 0,
√
and
dy dU ≥ 0 on Γ where y < 0 and |dx/dy| > −K.
(B)
dy dU ≤ 0√ for y ≤ 0 on
characteristics, dx/dy = ± −K.
(C) U satisfies a strong maximum principle in y > 0.
The first two properties, (A) and (B), easily fall out of the definition of U .
Proof of (C). In y > 0, using (2.1), we find
1 K 0 (y)
(α Ux + βUy )
2 K(y)
−1/2
α = 1 + Uy KUx2 + Uy2
−1/2
β = KUx KUx2 + Uy2
K(y) Uxx + Uyy =
where
and
are both bounded and measurable. Since K > 0 we can use standard elliptic theory
to complete the proof. We are assuming here that K, u and any relevant boundary
have sufficient smoothness.
Uniqueness theorems can be derived from (A), (B) and (C). To prove uniqueness
we need just to prove that the corresponding homogeneous problem has only the
zero solution u ≡ 0.
We consider the so-called Frankl problem:
The solution u satisfies (2.1) in Ω where Ω is a domain (see Figure 1) bounded
by Γ0 + Γ1 + Γ− . Here
0 lies in y ≥ 0 and is convex. Γ1 is a curve satisfying the
Γ√
√
dx
>
−K, dx
Frankl condition dx
dy
dy ≤ 0, and Γ− is a characteristic dy = − −K.
The boundary condition is:
(2.2)
u=0
on Γ0 + Γ1 .
D
Γ0
Ω
y =0
Γ−
Figure 1
Γ1
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CATHLEEN SYNGE MORAWETZ
Figure 2
We assume u, K and ∂Ω are smooth.
Theorem 1. [12]
u≡0
in ∂Ω.
Proof. At D, the unique point on ∂Ω where y is maximum, we have Uy = −u2y ≤ 0.
Hence U does not have its maximum in Ω ∩ {y > 0} at D by the strong maximum
principle (C). By (A) and (C), U does not have its maximum in Ω ∩ {y > 0}. Thus
U has its maximum on y ≤ 0. By applying (A) for y ≤ 0 and (B) it is easily seen
that U must be identically constant, and thus by the definition of U along with the
boundary condition (2) we have u ≡ 0.
Of course much more general domains Ω can be constructed, but they do not
include the very simple domain of Figure 2. One can, however, assert that for an
arbitrary piecewise smooth elliptic boundary the solution space of the homogeneous
problem is finite dimensional.
3. The wave equation
The next application of Noether’s theorem is to the wave operator:
∂2
− ∆.
∂t2
This operator comes from the action of a Lagrangian that is invariant under
translation, stretching, rotation on a sphere and, most useful of all as it turns out,
under the so-called conformal transformation, which would have been better called
inversion.
It is a lot of work to get all the conservation laws by Noether’s theorem; they
were rarely first found that way, and not all of them are useful. It is easier to use
generators or multipliers or what Friedrichs called the abc method. For example,
we get conservation of energy by multiplying u by ut and integrating by parts.
Thus
1 2
ut + |∇u|2 t − div (ut ∇u) .
ut u =
2
(3.1)
=
VARIATIONS ON CONSERVATION LAWS
145
Integrating over a slab from 0 to an arbitrary time, one sees that the integral of
the energy density, 12 u2t + |∇u|2 , does not change in time if the solution vanishes
far out. This is the most used of the conservation laws.
Next comes the conservation law that we obtain under a conformal transformation. First we take the transformation,
x, t, u → X, T, U
|x| = r
and |X| = R,
(x, t) = (X, T )/(T 2 − R2 ) i.e.
(X, T ) = (x, t)/(t2 − r2 ),
and, with m the number of space dimensions,
r
m−1
2
u=R
m−1
2
U.
Then
u = Factor
× U.
m−1
m−1
The easy way to check this invariance is to set r 2 u = R 2 U = W , t + r =
1/(T − R) = ξ = 1/Ξ, and t − r = 1/(T + R) = η = 1/H.
In the end, we obtain
(3.2)
(t2 + |x|2 ) ut + 2tx · ∇u + (m − 1) u u = Q + divP,
which is just the energy identity for U .
The strength of this identity is that the contribution from any space-like surface,
for example, t = constant, has a definite sign because the image of the surface in
(X, T ) space is also space-like.
The first example [13] of its use is for a solution of the wave equation in the
exterior of a star-shaped obstacle, on which the solution vanishes. Integrating by
parts, again in the slab (0 < t < t1 ), one obtains a bound for the integral on t = t1
in terms of the initial data. The term from Q can be rewritten as integrals with
positive integrands. Thus,
Z
Z
Z
= qdx dy dz P · n dS + q dx dy dz t=T
t=0
where q = q1 + q2 + q3 and
q1 = (r2 + t2 )(|∇u|2 − u2r )
1
2
(r + t)2 ((ru)r + r ut )
q2 =
4r
1
2
(r − t)2 ((ru)r − r ut )
q3 =
4r
and the term in P is
Z
t
∂u
∂n
2
~x · ~n dS dt
over the obstacle. Here ~n is the normal derivative. Star-shaped means ~x · ~n ≥ 0.
Every term is nonnegative, so each is bounded, and thus the first term yields
the decay of the L2 space norm of the angular derivatives, in fact, like l/t. The
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CATHLEEN SYNGE MORAWETZ
second shows that the L2 space norm of the derivative in the outward characteristic direction also decays like l/t. These ultimately yield Sommerfeld’s radiation
condition.
The remaining derivative in the inward direction decays like l/t only in a local
L2 norm.
That is the best, there are in 2D for obstacles that do not trap rays. In 3D,
Huyghens principle will give you exponential decay, but that is a long story; see
the book by Lax and Phillips [8].
Not surprisingly, there are analogous formulas for the reduced wave equation in
the exterior of a star-shaped obstacle. These give estimates for the error created by
high-frequency approximation [10] (geometrical optics). The formulas were found
by hunting and fishing for a workable multiplier analogous to the multiplier of (3.2).
Decay of a similar kind can also be proved for semilinear equations of the form:
(3.3)
u + F 0 (u) = 0
with F 0 (u) = d F (u)/d u satisfying some conditions beyond making the energy
contribution from F 0 (u) positive, that is, F (u) ≥ 0.
Of course the conformal transformation does not lead to a conservation law. The
extra terms in the previous divergence equation are
(3.4)
(t2 + r2 ) F (u) t + t ((m − 1) u F 0 (u) − 2(m + 1) F ) .
To get decay, the second term, which gets integrated over space and time, must be
nonnegative, which forces F to satisfy
(m − 1) u F 0 (u) − 2(m + 1) F ≥ 0.
As a result you cannot get decay if, say, F (u) = u2 + u4 , m = 3, the semilinear
example, the nonlinear Klein-Gordon equation, first proposed by Heisenberg. For
that case a very weak decay can be obtained, with a different multiplier [14], that
can be jacked up eventually to give a scattering theory for the equation [15]. A
scattering theory tells us the relationship between the solution at t = −∞ and the
solution at t = +∞.
Strichartz [21], followed by Ginibre and Velo [5], altered the art of finding decay
for semilinear equations, obtaining uniform Sobolev space estimates in many cases.
So multipliers and invariants do not have all the answers.
On the other hand, conservation identities that were used originally for decay
provide an unusual and useful handle for some otherwise unobtainable local estimates, as Shatah and Struwe have done in their studies of regularity for semilinear
problems in the so-called critical case, critical because standard Sobolev inequalities
do not help. Thus
(3.5)
F 0 (u) = u|u|p−2
with p = 2m/(m − 2). See also Bourgain [1] for the corresponding nonlinear
Schrödinger equation.
We get the flavor of their method in the critical case and consider u real and
positive for convenience.
Here is the problem. We can solve (3.3), (3.5) for short times as a C ∞ solution
if the Cauchy data on t = 0 are C ∞ . But does the solution remain so regular for
all times? If not, there is a first time t0 when the regularity ceases. This could be
at many different points in space. Let one of them be the origin. The “solution”
at (0, t0 ) depends on the characteristic cone drawn backwards in time, and it is
VARIATIONS ON CONSERVATION LAWS
147
t = t0
=0
t = −ε
M
TC
t = −T
D
TC : truncated cone
< T,
D : | x| _
t =−T
M : t + |x| = 0 , −T ≤ t ≤ −ε ≤ 0
Figure 3
from properties in that cone combined with the multiplier method that we find
a very minimal regularity inside the cone as we approach (0, t0 ). That minimal
regularity is described as follows; see Figure 3. Let C be the backward half cone
t0 − t > |x − x0 |, and let M be its mantel t0 − t = |x − x0 |. We slice the cone into
a truncated cone, T C, by t0 − t = T and t0 − t = where T > 0, and will be
smaller than anything. The standard energy identity can be applied to T C since
the solution is regular for t < t0 .
Take t0 = 0. Then (a subscript i on u means a derivative),
0 = ut utt − uii + up−1
p
2
u u u
ut
i i
− (ut ui )i +
+
,
=
2 t
2 t
p t
or integrating over the cone,
p Z 2
u
ut
ui ui
tn − ut ui xin +
tn +
dS
0=
2
2
p
∂Ω
where ui xin = ur rn on the mantel,
2 !
Z
|∇u|2 − u2r
ut − ur
up dSe
√
+
dSe +
=
2
2
p
2
mantel of TC
Z t=
up
1 2
ut + |∇u|2 +
|dx|
+
2
p
t=T
where dSe is an appropriate surface element on the cone r + t = 0.
Thus not only is the energy bounded by its initial value, but the flux across
the mantel of T C which is monotonically increasing is also uniformly bounded
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CATHLEEN SYNGE MORAWETZ
and hence has a limit. Thus we can turn the argument around and say that
for T sufficiently small, the flux across the mantel of T C is o(1) in terms of the
energy. We note for further use that if this expression is rewritten in terms of
m−1
w = rR 2 u, one can integrate the indefinite terms in w wr where r = |x| and find
that mantel of T C (wt − wr )2 dr dσ is also o(1) where dr dσ is the surface element in
polar coordinates on the cone. This can easily be recognized from the spherically
m−1
symmetric case. Using w = r 2 uRand Schwarz’ inequality on the flux form in
w and the flux form in u, one finds u2 r−2 dS is also o(1). The energy on t = is, however, by this kind of argument only bounded. Some of the energy may vent
through the tip of the cone, and we cannot eliminate this by energy consideration
alone. Further
Z
up
−m+1
2
2
2
2
wt + wr + |∇u| − ur +
|dx|
r
p
t=−
is bounded by the energy at t = T .
The precise result we are after is that
Z
(3.6)
up |dx| = 0.
lim
T →0
|x|<−T
This is a very weak result. But Shatah and Struwe [20] show that it can be jacked
up to show that the solution can be continued as a regular solution beyond 0, in
particular in the critical case.
To prove (3.6), we use an estimate obtained by employing the multiplier1 t ut +
u.
x · ∇ u + m−1
2
The easiest way to obtain the most useful identity is to compute it the oldfashioned way with subscripts. However, before plunging in, we point out that a
more elegant way is to notice that a shift in t would take us from the multipliers
u and that the identity we obtain
of (3.2) to the multiplier x · ∇u + t ut + m−1
2
must be the same as the identity obtained by shifting t in the old identity. Then
we would collect terms to get definite expressions. Here we work directly. Thus
m−1
u utt − uii + up−1
0 = t ut + xj uj +
2
= Qt + P ii + R
with
1 2 1
m−1
tup
t ut + t ui ui + xj uj ut +
u ut +
2
2
2
p
1
1
m
−1
1
uui + up xi
P i = −t ut ui − xi u2i − xj uj ui + xj uj uj −
2
2
2
ρ
(1 + m) m − 1
+
R= −
up .
p
2
Q=
We note that in the critical case R ≥ 0.
1 This multiplier was used in finding decay of energy near an obstacle [13] and not as it is
frequently referenced in the paper [14] on the Klein-Gordon equation.
VARIATIONS ON CONSERVATION LAWS
149
The next step is a slightly tricky rearrangement to absorb the indefinite terms
m−1
uui using xirui = ur and w = r 2 u; then
like m−1
2
1
Q = t |∇u|2 − u2r
2
1
m−1
t up
1
uut +
+ (t + r)(ut + ur )2 + (t − r)(ut − ur )2 +
4
4
2
p
1
= t |∇u|2 − u2r
2
1
t up
1
.
+ (t + r) r−m+1 (wt + wr )2 + (t − r) r−m+1 (wt − wr )2 +
4
4
p
We integrate over the truncated cone T C. It is now quite easy to see from the
second form of Q that we may let → 0 using the energy bounds of the previous
argument.
The rest of the argument follows by showing that the integral over the mantel is
T o(1). This is because the integral over the base, t = T , has the same sign as the
volume term (second form of Q); hence the
R integral over the base is T o(1).
R Further,
each term in Q is nonnegative. Hence T up |dx| is T o(1), and thus up |dx| is
o(1).
All that is left is to check that the integral over the mantel is T o(1). The integral
over the mantel where dt = −dr is:
Z xi drdσ
Q + Pi
r
where again drdσ is a surface element. We note that
1
1
m−1
r
xi
= −t ut ur − r u2t − r u2r + r |∇u|2 −
u ur + up
Pi
r
2
2
2
p
t 2
1
m−1
t
ut − u2r − t ut ur −
u ur − up .
= − t |∇u|2 − u2r +
2
2
2
p
R
P i xi
dr dσ, by using the bounds on the flux across the cone in
Thus mantel Q + r
terms of u, is T o(1).
We have utilized here the bound on the integral u2 /r2 on the cone that the
p
second energy form gives. Note that the term in up and the term involving ut + ur
drop out.
This completes the argument. Jacking the result up to full regularity is complicated, and we do not try to do that here.
4. Black holes
I turn now to my last topic, black holes. What does that have to do with
conservation laws and the wave equation? They are related, but frankly this last
topic came about because when all the world was telling me what to talk about
on this occasion, someone said, “black holes.” And I thought, “It doesn’t fit my
title, and I have forgotten what they really are.” But I went home and looked in
the encyclopedia which sits in our living room. (It is very beneficial for resolving
domestic disputes to have an encyclopedia in the living room.) I looked up “black
holes”. There I learned of the three ways in which a star can die.
Most stars are composed of hydrogen. But let me digress a moment. That most
stars are hydrogen was discovered by Cecilia Payne [18] through spectral analysis
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CATHLEEN SYNGE MORAWETZ
at Harvard in 1923, but the big-shot astronomers pooh-poohed it and talked her
out of her convictions. She wrote her thesis up as a “maybe” and frankly did not
get the credit until long after the truth had become evident.
But back to the death of stars. Most stars make energy by converting their
hydrogen into helium and down the periodic table into other elements. The first
kind of star is small, the process stops and the star becomes a “white dwarf”.
“Small” means having a mass like our sun’s. The second kind of star, postulated
by Chandrasekhar and much bigger, becomes a “supernova” and explodes its mass
into the universe. The pressure beats the gravitational pull. The third kind of star
(about fifty solar masses) turns into iron, and there is nowhere to go. Further fusion
would not release more energy. Again the pressure of the gas competes with the
gravitational pull to implode. The speeds and forces tell us that the phenomenon
is to be treated by general relativity, and that is where the wave operator comes in.
The bending and twisting of the characteristics of the system give the gravitational field and, despite the countervailing pressure of the gas, create a black hole
into which everything falls and from which no signals emerge. It appears to have
been so named by John Wheeler, and the name alone excites everyone’s fancy.
That is more or less what the encyclopedia said. Looking for more, I moved
on to a popular book, Black Holes and Time Warps, by Kip Thorne [23]. I was
reminded of Alice in Wonderland.
“What is the use of a book,” thought Alice, “without pictures or conversation?”
Thorne’s book had plenty of pictures and conversation. In fact, it’s very good
reading. But this Alice is a mathematician and my thought was, “What is the
use of a book in science without equations or formulas?” So I turned to more
mathematical sources, mostly friends. I particularly want to acknowledge my late
father, John L. Synge; Martin Kruskal; Satya Kichenessamy; Jalal Shatah; and
Demetrious Christodoulou.
The problem of the black hole actually goes back to the end of the eighteenth
century, when many believed that light was composed of corpuscles of mass subject
to gravity. John Michell [11] (like many other rectors in the Church of England, he
preached on Sunday and did science the rest of the week) determined the radius
of a star from which light cannot leave because the velocity of light (known with
some accuracy at the time) is less than the escape velocity. Remember, that is the
velocity a vehicle needs to be able to escape the gravitational field of a star. Stars
from which light could not escape were called “dark stars”. That was two hundred
years before relativity.
The first relativistic model for a dark star was proposed by Schwarzschild [19]
in 1916 in two papers written on the Eastern front, where he was serving in the
German army. (He died of illness soon after.) His model was imperfect, and there
was a deep underlying problem.
But now this Alice is sinning! We need some equations and formulas. We begin
with the metric in space-time:
The Einstein metric is given by
{gα β } = G,
ds2 = gα β dxα dxβ
−1
Locally G can be written as
0
0
1
1
.
1
α β = 0, 1, 2, 3.
VARIATIONS ON CONSERVATION LAWS
151
The Einstein equations are
E(G) = T.
The left hand side, E(G), is geometric; it is associated with curvature in space-time.
T is a tensor which governs the gas dynamics. The operator E may be written as
E = G + Q1 (G) + Q2 (G).
At last we see the wave operator, followed by Q1 and Q2 , which are nonlinear.
They depend on G and its first two derivatives. Q1 is quasilinear—that is, it is
linear in the second derivatives of G—and Q2 is quadratic in the first derivatives.
In the Einstein equations, geometry is separated from the gas dynamics, but
there are a lot of unknowns. This is already a mess, and there are also compatibility
conditions. But as it turns out the count is right for the system of p.d.e.
We pass over the famous work on the initial value problem by Yvonne Choquet
Bruhat [2] and the more recent work of Christodoulou and Klainerman [4] and
many others.
What are we looking for when we look for a black hole? For a black hole we
need a solution to these equations, that is, a metric plus gas variables (we do have
to put in some gas laws) that form a solution in the weak sense starting from some
initial manifold where we may say these equations take over from some sort of
fiery boundary layer. But this solution must FAIL at some later time—that is the
HOLE. All particle paths must fall into it. To be BLACK we have to be able to
say we cannot see into it; that is, we cannot receive a signal from it.
Let us try for a simple case — spherical symmetry. The metric will then have
the form:
ds2 = g11 dr2 + 2g12 dr dt + g22 dt2 + r2 dθ2 + sin2 θ dφ2 .
Case 1. The star is a point mass. The metric is the Schwarzchild metric:
−1
2m
2m
2
2
dr − 1 −
dt2 + r2 dθ2 + sin2 θ dφ2 .
ds = 1 −
r
r
This is certainly expected to describe the metric at large distances.
Here m is the mass of the star. There are two singularities:
r = 2m,
r = 0.
It is clearly all right to have a singularity at r = 0, but what about r = 2m? It
turns out that this is not a real singularity and can be fixed by making a change of
the independent variables.
My father, John L. Synge [22], in a paper presented to the Royal Irish Academy in
1949, following Lemaı̂tre [9] and H.P. Robertson, constructed a change of variables
that continued the metric smoothly and where a full description could be made of
the paths of particles in space-time not only near r = 2m but also near the true
singular point r = 0. A better change of variables by Martin Kruskal [7]—that is,
a better mapping—fixed the remaining defects of the Schwarzschild metric.
As a last stop on the black hole story, I would like to describe the results of the
model of Christodoulou [3], who began this work more than 10 years ago. He uses
a gas with a somewhat strange gas law. It is due to Oppenheimer [17] and has been
152
CATHLEEN SYNGE MORAWETZ
used by Bethe and others. Don’t forget this gas is a very unusual end product of a
long process.
Again under spherical symmetry, the metric may be put in the form
−Ω2 du dv + r2 dθ2 + sin2 θ dφ2 .
u = constant represents a light ray moving out with fixed latitude and longitude;
v = constant correspondingly moving in.
We have to find r and Ω as functions of u and v. It is easier to work with the
mass-energy density m instead of Ω. But they are simply related:
2m
= −4ru rv .
Ω 1−
r
2
The equations reduce to
−1
2m
ru rv ,
r ruv = (2m/r) 1 −
r
2m 2 2
r Φu ,
ru mu = 2π 1 −
r
2m 2 2
r Φv ,
rv mv = 2π 1 −
r
r Φuv + ru Φu + rv Φu = 0.
Note the wave operator appearing here in characteristic form. And also note that
the equations are clearly singular for r = 0 and for r = 2m. But again the results
show that only r = 0 is a genuine singularity.
Besides r and m we now have Φ. This comes from the gas model. It makes
it appear that there are 4 equations for 3 unknowns, but computation shows that
the last equation is the compatibility condition that m satisfy the two preceding
ordinary differential equations on the characteristics.
Christodoulou [3] has proved the existence of solutions of bounded variation
that are unique. Data is specified for m and r on u = 0, let us say, and because of
spherical symmetry on u − v = 0.
A rather mild condition must be imposed on the data in order to end up with a
black hole.
Christodoulou makes the figure (see Figure 4) with the u, v axes rotated through
450 .
Along with the u, v space there is the spherical coordinate space θ, φ, and so we
are only interested in the right half plane and have r = m = 0 on the axis.
There is a singularity for all of these solutions where r = 0, m > 0. Various
derivatives blow up. This is the HOLE.
There is a second curve, A, where r = 2m. The solution can be continued across
it, but every world path then proceeds into the hole. It forms what Christodoulou
calls an apparent horizon. Notably it is asymptotic as v → ∞ to a characteristic (a
light ray) u = constant. That is H. It is really a cone in the 4-space. It contains
the paths of the last signals that can go off to ∞. This makes the hole BLACK.
No light paths escape beyond it to ∞.
VARIATIONS ON CONSERVATION LAWS
u
B0
153
B
A
H
v
u −v = 0
Figure 4
Much more could be said about the mathematics of black holes. Physicists have
explored the interaction of, say, two black holes. Where does that fit in? And so
on.
But where are the conservation laws, and where are the invariants?
The equations are invariant under a change of scale, but they do not seem to
have any conservation laws (nor should they). In fact, the change of scale invariance
makes it harder to make estimates instead of easier.
The space of Bounded Total Variation comes naturally from 1D-time gas dynamics, but it is not right there for more space variables, so we do not expect it
to be right here if we abandon spherical symmetry. We know L2 estimates are no
good for these problems even for small data.
So here is the wave equation—in desperate need of new estimates, for the world
is not likely to be spherically symmetric. And that is where we will leave it. Black
holes bring us to the wave operator, but not to its conservation laws.
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